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Invariant manifolds for physical and chemical kinetics
Abstract
Attached is an unedited file of the book, before Springer editorial preparation.
The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the constructive methods of invariant manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space ({\it the invariance equation}). The equation of motion for immersed manifolds is obtained ({\it the film extension of the dynamics}). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus {\it slowness is presented as stability}.
A collection of methods to derive analytically and to compute numerically the slow invariant
manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn ~ 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; kinetic theory of phonons; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, cell division kinetics.
... Both levels are autonomous, both have been developed independently on the basis their own family of experimental observations, and both can be also applied independently. In addition, the experimentally observed behavior of most macroscopic systems can also be described by a sequence of intermediate (mesoscopic) autonomous levels, as for instance the level of kinetic theory and the level of hydrodynamics [7,23]. ...
... In the latter pattern recognition process, we search for fixed points (that are the vector fields generating the reduced time evolution). The first viewpoint of reductions follows the pioneering works of Chapman and Enskog [8] in their investigation of the reduction of the Boltzmann kinetic equation to hydrodynamics (see also [23]). The second viewpoint of reductions follows Grad [25] in his investigation (that begins with a hierarchy reformulation of the Boltzmann kinetic equation) of the same physical problem. ...
... The first, that we shall call Chapman-Enskog-type reductions, follow the pioneer investigation [8] More specifically, the manifold M ⊂ M is sought as an appropriate deformation (see e.g. [23], [47], [67]) of the dissipation equilibrium manifold M (diss) ⊂ M , that is, in the case of the Boltzmann kinetic theory, composed of local Maxwellian distribution functions (see Section 2.1). ...
The General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) provides the structure of mesoscopic multiscale dynamics that guarantees the emergence of equilibrium states. Similarly, a lift of the GENERIC structure to iterated cotangent bundles, called a rate GENERIC, guarantees the emergence of the vector fields that generate the approach to equilibrium. Moreover, the rate GENERIC structure also extends Onsager’s variational principle. The maximum entropy principle in the GENERIC structure becomes the Onsager variational principle in the rate GENERIC structure. In the absence of external forces, the rate entropy is a potential that is closely related to the entropy production. In the presence of external forces when the entropy does not exist, the rate entropy still exists. While the entropy at the conclusion of the GENERIC time evolution gives rise to equilibrium thermodynamics, the rate entropy at the conclusion of the rate GENERIC time evolution gives rise to rate thermodynamics. Both GENERIC and rate GENERIC structures are put into the geometrical framework in the first paper of this series. The rate GENERIC is also shown to be related to Grad’s hierarchy analysis of reductions of the Boltzmann equation. Chemical kinetics and kinetic theory provide illustrative examples. We introduce rate GENERIC extensions (and thus also Onsager-variational-principle formulations) of both chemical kinetics and the Boltzmann kinetic theory.
... These reversible reactions are called fast reactions. After reaching equilibrium, the fast reactions make no contribution to the evolution of concentration for each reactant [22]. By introducing a small parameter 0 < ϵ ≪ 1 to character the fastness of these reactions, which equals to the ratio of relaxation times ...
... . Quasi-steady-state approximation In contrast to rapid balance assumption on reversible reactions by PEA, QSSA assumes that the production and consumption rates of some species (saying, the first B species) are equal, so that they will stay in a quasi-steady state after a relatively short period of time[22]. We still adopt the same notation, a small parameter 0 < ϵ ≪ 1 to characterize the difference in the relaxation time between the first B species and the remaining (N − B) species. ...
Partial equilibrium approximation (PEA) and quasi-steady-state approximation (QSSA) are two classical methods for reducing complex macroscopic chemical reactions into simple computable ones. Previous studies mainly focus on the accuracy of solutions before and after applying model reduction. While, in this paper we start from a thermodynamic view, and try to establish a quantitative connection on the essential thermodynamic quantities, like entropy production rate, free energy dissipation rate and entropy flow rate, between the original reversible chemical mass-action equations and the reduced models by either PEA or QSSA. Our results reveal that the PEA and QSSA do not necessarily preserve the nice thermodynamic structure of the original full model during the reduction procedure (e.g. the loss of non-negativity of free energy dissipation rate), especially when adopting the algebraic relations in replace of differential equations. These results are further validated though the application to Michaelis-Menten reactions analytically and numerically as a prototype. We expect our study would motivate a re-examination on the effectiveness of various model reduction or approximation methods from a new perspective of non-equilibrium thermodynamics.
... In addition to more classical classifiers, we have attempted the construction of maximum Shannon entropy-based probabilistic classifier based on the concept of Quasi Equilibrium Manifold as defined in [29,30] and implemented in the discrete form of Quasi Equilibrium Grid (QEG) as discussed in [31,32,33,34]. The main idea is described below. ...
... In our study, we made use of known classifiers. In addition, we also investigate a Bayesian type classifier based on the concept of Quasi-Equilibrium Manifold [29,30,31,33], as detailed above. ...
... In addition to more classical classifiers, we have attempted the construction of maximum Shannon entropy-based probabilistic classifier based on the concept of Quasi Equilibrium Manifold as defined in [29,30] and implemented in the discrete form of Quasi Equilibrium Grid (QEG) as discussed in [31,32,33,34]. The main idea is described below. ...
... In our study, we made use of known classifiers. In addition, we also investigate a Bayesian type classifier based on the concept of Quasi-Equilibrium Manifold [29,30,31,33], as detailed above. ...
In this study, we evaluate several classifiers and focus on selecting a minimal set of appropriate material features. Our objective is to propose and discuss general strategies for reducing the number of descriptors required for material classification. The first strategy involves testing whether the critical temperature of the target material property is invariant with respect to binary groups of composition-based features. We also propose a multi-objective optimization procedure to reduce the set of composition-based material descriptors. The latter procedure is found to be particularly useful when applied to Bayesian classifiers. We test the proposed strategies focusing on low-temperature superconductors material data extracted from a public database.
... Invariance defect of the local Maxwell-Boltzmann distribution. The sum of the two above expressions, (147) and (148), is so important that it deserves a name of its own. We call it the defect of invariance of the local Maxwell-Boltzmann distribution function with respect to the Boltzmann equation, ...
... In an attempt to provide a unified an numerically efficient kinetic framework for non-ideal fluids simulation in [146] the authors introduced a projector K onto a local equilibrium at constant temperature [147], ...
This contribution presents a comprehensive overview of of lattice Boltzmann models for non-ideal fluids, covering both theoretical concepts at both kinetic and macroscopic levels and more practical discussion of numerical nature. In that context, elements of kinetic theory of ideal gases are presented and discussed at length. Then a detailed discussion of the lattice Boltzmann method for ideal gases from discretization to Galilean invariance issues and different collision models along with their effect on stability and consistency at the hydrodynamic level is presented. Extension to non-ideal fluids is then introduced in the context of the kinetic theory of gases along with the corresponding thermodynamics at the macroscopic level, i.e. the van der Waals fluid, followed by an overview of different lattice Boltzmann based models for non-ideal fluids. After an in-depth discussion of different well-known issues and artifacts and corresponding solutions, the article finishes with a brief discussion on most recent applications of such models and extensions proposed in the literature towards non-isothermal and multi-component flows.
... The former follows the pioneering work of Kirkwood [3,4] (that gave rise to the modern theoretical rheology [5]) where the internal structure enters on the level of kinetic theory and another pioneering work of Cosserat brothers [6] where the internal structure enters on the level of continuum theory [7][8][9]. The second way of physically motivated extension is inspired by Grad's reformulation [1,10,11]) of the Boltzmann kinetic equation (that gave rise to extended thermodynamics [12,13] and extended irreversible thermodynamics [14]). ...
Approach of mesoscopic state variables to time independent equilibrium sates (zero law of thermodynamics) gives birth to the classical equilibrium thermodynamics. Approach of fluxes and forces to fixed points (equilibrium fluxes and forces) that drive reduced mesoscopic dynamics gives birth to the rate thermodynamics that is applicable to driven systems. We formulate the rate thermodynamics and dynamics, investigate its relation to the classical thermodynamics, to extensions involving more details, to the hierarchy reformulations of dynamical theories, and to the Onsager variational principle. We also compare thermodynamic and dynamic critical behavior observed in closed and open systems. Dynamics and thermodynamics of the van der Waals gas provides an illustration.
... When a chemical reaction has many species participating in the current reactions, these reductions are made by keeping slow and fast phases in account [13,14]. In a dissipative reaction system, all possible solution curves or trajectories gradually make hyperplanes and reach the equilibrium point [15]. ...
This research explores the intricate concept of the Slow Invariant Manifold (SIM) and its pivotal role in developing model reduction techniques (MRTs) for challenges within dissipative systems in chemical kinetics, specifically in mechanical engineering. Focusing on the multi-step mechanism with two intermediates, primary approximations of the SIM are constructed and compared using two prominent MRTs: The Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). At the given rate coefficient, a special computational experiment was performed in which the efficiency of chemical species has been compared. Noteworthy innovation involves evaluating SIM separately for reduced species, departing from the conventional approach of considering every species within the mechanism. The study employs local sensitivity analysis with MATLAB's Sim-Biology toolbox, presenting quantitative findings in a tabular format for a comprehensive MRT comparison. Beyond contributing to a deeper understanding of model reduction techniques in complex chemical kinetics, this research marks the first computational exploration of dissipative systems. The novel perspective of evaluating SIM for reduced species offers nuanced insights, emphasizing the critical role of SIM in effectively addressing challenges in mechanical engineering applications. In summary, this study introduces computational advancements and novel approaches, advancing model reduction techniques and highlighting the significance of SIM in addressing challenges within mechanical engineering contexts.
... The induced dynamics on the parameter space are described by the vector field dy/dt which is equal to (1 − P). J(x) is the invariance defect, and the fast plain for the affine subspace x + kerP [15]. ...
A comprehensive comprehension of the concept of the gradual unchanging manifold (GUM) gives rise to numerous techniques for reducing models (TRM) in dissipative systems of chemical kinetics for mechanical engineering problems. This research calculates and contrasts the unchanging solutions for multi-route systems. The TRM, such as the Intrinsic Low Dimensional (ILDM), Quasi Equilibrium Manifold (QEM), and Spectral Quasi Equilibrium Manifold (SQEM), are utilized to construct the gradual unchanging manifolds, which function as primary approximations to GUM. By employing the Method of Invariant Grid (MIG), the enhancements of QEM solutions are computed imposing the effect of activation energy. Our discoveries suggest that we could assess each route independently rather than taking into account the entire mechanism. Local sensitivity analysis is conducted using the SimBiology toolbox of MATLAB. To compare TRM, the calculation period is presented in tabular form.
We present “on the fly” algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in Goussis (Combust Theor Model 16:869–926, 2012) that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis–Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models.
We propose a novel nonlinear manifold learning from snapshot data and demonstrate its superiority over proper orthogonal decomposition (POD) for shedding-dominated shear flows. Key enablers are isometric feature mapping, Isomap, as encoder and, $K$ -nearest neighbours ( $K$ NN) algorithm as decoder. The proposed technique is applied to numerical and experimental datasets including the fluidic pinball, a swirling jet and the wake behind a couple of tandem cylinders. Analysing the fluidic pinball, the manifold is able to describe the pitchfork bifurcation and the chaotic regime with only three feature coordinates. These coordinates are linked to the vortex-shedding phases and the force coefficients. The manifold coordinates of the swirling jet are comparable to the POD mode amplitudes, yet allow for a more distinct and less noise-sensitive manifold identification. A similar observation is made for the wake of two tandem cylinders. The tandem cylinders are aligned and located at a streamwise distance which corresponds to the transition between the single bluff body and the reattachment regimes of vortex shedding. Isomap unveils these two shedding regimes while the Lissajous plot of the first two POD mode amplitudes features a single circle. The reconstruction error of the manifold model is small compared with the fluctuation level, indicating that the low embedding dimensions contain the coherent structure dynamics. The proposed Isomap– $K$ NN manifold learner is expected to be of great importance in estimation, dynamic modelling and control for a large range of configurations with dominant coherent structures.
In the last decade, minimal kinetic models, and primarily the lattice Boltzmann equation, have met with significant success in the simulation of complex hydrodynamic phenomena, ranging from slow flows in grossly irregular geometries to fully developed turbulence, to flows with dynamic phase transitions. Besides their practical value as efficient computational tools for the dynamics of complex systems, these minimal models may also represent a new conceptual paradigm in modern computational statistical mechanics: instead of proceeding bottom-up from the underlying microdynamic systems, these minimal kinetic models are built top-down starting from the macroscopic target equations. This procedure can provide dramatic advantages, provided the essential physics is not lost along the way. For dissipative systems, one essential requirement is compliance with the second law of thermodynamics. In this Colloquium, the authors present a chronological survey of the main ideas behind the lattice Boltzmann method, with special focus on the role played by the H theorem in enforcing compliance of the method with macroscopic evolutionary constraints (the second law) as well as in serving as a numerically stable computational tool for fluid flows and other dissipative systems out of equilibrium.
Summary: The task of constructing higher-dimensional invariant manifolds for dynamical systems can be computationally expensive. We demonstrate that this problem can be locally reduced to solving a system of quasilinear PDEs, which can be efficiently solved in an Eulerian framework. We construct a fast numerical method for solving the resulting system of discretized nonlinear equations. The efficiency stems from decoupling the system and ordering the computations to take advantage of the direction of information flow. We illustrate our approach by constructing two-dimensional invariant manifolds of hyperbolic equilibria in $Bbb R^3$ and $Bbb R^4$.
The kinetic theory of gases, including Granular Gases, is based on the Boltzmann equation. Many properties of the gas, from
the characteristics of the velocity distribution function to the transport coefficients may be expressed in terms of functions
of the collision integral which we call kinetic integrals. Although the evaluation of these functions is conceptually straightforward,
technically it is frequently rather cumbersome. We report here a method for the analytical evaluation of kinetic integrals
using computer algebra. We apply this method for the computation of some properties of Granular Gases, ranging from the moments
of the velocity distribution function to the transport coefficients. For their technical complexity most of these quantities
cannot be computed manually.
This article is concerned with
the asymptotic accuracy of the
Computational Singular Perturbation (CSP)
method developed by Lam and Goussis [The CSP method for simplifying kinetics,
Int. J. Chem. Kin. 26 (1994) 461–486]
to reduce the dimensionality of a
system of chemical kinetics equations.
The method,
which is generally applicable to
multiple-time scale problems
arising in a broad array of scientific disciplines,
exploits the presence of
disparate time scales to model
the dynamics by an evolution equation
on a lower-dimensional slow manifold.
In this article it is shown that
the successive applications
of the CSP algorithm generate,
order by order,
the asymptotic expansion
of a slow manifold.
The results are illustrated
on the Michaelis–Menten–Henri
equations of enzyme kinetics.
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Padé approximants, and invariance principle are compared both in linear and nonlinear situations.
We discuss a new approach to nonequilibrium statistical thermodynamics based on mappings of the microscopic dynamics into the macro-scopic dynamics. Near stationary solutions, this mapping results in a compact formula for the macroscopic vector field without a hypothesis of a separation of time scales. Relations of this formula to short-memory approximation, the Green-Kubo formula, and expressions of transport coefficients in terms of Lyapunov exponents are discussed. Se discute una nueva aproximación a la termodinámica estadística fuera de equilibrio basándose en mapeos de la dinámica microscópica dentro de la dinámica macroscópica. En soluciones casi estacionarias, este mapeo da lugar a una fórmula compacta para el campo vectorial macroscópico, sin una hipótesis de separación de escalas de tiempo. Se discuten las relaciones de esta fórmula con la aproximación de memoria corta, la fórmula de Green-Kubo y las expresiones de los coeficientes de transporte en términos de los exponentes de Lyapunov. Descriptores: Mecánica estadística fuera de equilibrio; relación de fluctuación-disipación exacta.
The nonlinear recurrence Chapman–Enskog solution for the linearized Grad ten-moment equations is resummed exactly. Using this solution, the stability of higher-order hydrodynamics in various approximations is discussed.
In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem. Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium. We discuss performance of this approach as compared to the standard realizations.
Suppose that the slow invariant manifold is found for a dissipative system. What have we constructed it for? First of all, for solving the Cauchy problem, in order to separate motions. This means that the Cauchy problem is divided in the following two subproblems:
In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed (the method of invariant grids). We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appear only in the calculation of the tangent spaces. This fact determines high computational efficiency of the method of invariant grids. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis–Menten mechanism), and the hydrogen oxidation. The algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.